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54: General topology


Introduction

Topology is the study of sets on which one has a notion of "closeness" -- enough to decide which functions defined on it are continuous. Thus it is a kind of generalized geometry (we are still interested in spheres and cubes, for example, but we might consider them to be "the same", yet distinct from a bicycle tire, which has a "hole") or a kind of generalized analysis (we might think of the functions f(x)=x^2 and f(x)=|x| as being "the same", and yet distinct from f(x)=signum(x)=x/|x|, which has a discontinuity).

More formally, a topological space is a set X on which we have a topology -- a collection of subsets of X which we call the "open" subsets of X. The only requirements are that both X itself and the empty subset must be among the open sets, that all unions of open sets are open, and that the intersection of two open sets be open.

This definition is arranged to meet the intent of the opening paragraph. However, stated in this generality, topological spaces can be quite bizarre; for example, in most other disciplines of mathematics, the only topologies on finite sets are the discrete topologies (all subsets are open), but the definition permits many others. Thus a general theme in topology is to test the extent to which the axioms force the kind of structure one expects to use and then, as appropriate, introduce other axioms so as to better match the intended application.

For example, a single point need not be a closed set in a topology. Does this seem "inappropriate"? Then perhaps you are envisioning a special kind of topological space, say a a metric space. This alone still need not imply the space looks enough like the shapes you may have seen in a textbook; if you really prefer to understand those shapes, you need to add the axioms of a manifold, perhaps. Many such levels of generality are possible.

Since the axioms of topology are stated in terms of subsets of X, it should be no surprise that one branch of topology is closely related to set theory, particularly "descriptive set theory". Here one considers general constructs such as closures of sets, limits, convergence, and nets. One can look at topologies related to order or cardinality, and so create extraordinarily large topological spaces. By using the axiom of choice, one may prove the existence of topological spaces with peculiar properties. In particular, there are questions about topology which can be reduced to questions of set theory, whose answer then depends on the axioms of set theory chosen.

As in other branches of axiomatic mathematics, we may (in category-theoretic style) make some basic constructs. The most important functions between topological spaces are the continuous ones (a definition borrowed from analysis), which we use to define homeomorphisms -- functions which can be used to demonstrate that two spaces are "the same". We can define products of spaces (and coproducts -- unions), subspaces, quotient spaces, and so on. On the one hand, each is a "universal" solution to some problem which can be stated in terms of the existence of maps and spaces related via commutative diagrams; this aspect makes them useful tools for algebraic topology. On the other hand, each can be studied (and generalized) internally, making them useful tools for analysis (including semicontinuous functions for example).

In order to get more significant results, one must restrict to spaces with some additional properties. The precise set of additional axioms depends on the intended results. For example, if we would like to know which spaces might have a topology which is consistent with a metric, we know individual points must be closed; the axiom that this is true (the "T_1" axiom) is but one of a number of separation axioms. A great deal of work has been done to see the independence of these axioms, the extent to which they are preserved under the constructions of the previous paragraph, and so on. The same is true of other types of axioms designed to focus attention on "well-behaved" spaces. For example, there are cardinality axioms (e.g., metric spaces have the additional property that the topology at a point is countable), compactness axioms (e.g. a space would have to be locally compact to be a manifold), connectedness axioms, and so on. In each case, there are a number of choices for how tight the axioms should be: is one interested in weak conclusions about a large family of spaces, or stronger conclusions about a family of more particular interest? Well-known results concerning these properties include a version of the Baire Category Theorem (nowhere dense subsets), Tychonoff's theorem (products of compact spaces), Urysohn's Lemma and Tietze's Theorem (functions on well-separated spaces), and compactness criteria (Bolzano-Weierstrass, Heine-Borel).

A number of families of spaces are defined by the presence of some extra structure which is related to the topology in a natural way. In each case, one can for example ask, given a topological space, whether the extra can be imposed on that space. We have already mentioned metric spaces: spaces on which there is a distance function; the latter question is then the question of whether a space is metrizable. Other categories include measure spaces (spaces with a given real-valued measure on families of subsets), manifolds (spaces with a given collection of coordinate charts), simplicial complexes (a generalization of polyhedra), CW-complexes (spaces with a given decomposition into subsets homeomorphic to balls of various dimensions), ordered topological spaces, topological groups or vector spaces, and so on. The distinction between this and the previous paragraph is that additional axioms are assumed about a new construct provided at the outset, rather than additional axioms about the topology; thus the questions asked about these structures can be about either the topology or about the new construct. (The discussion of these additional properties gives us subdisciplines Metric Topology, Combinatorial Topology, and so on.)

Moving toward applications, we can ask about topological spaces which arise in some fairly common way. One significant family of examples is sets S of functions between topological spaces X and Y. Depending on the properties or additional structures possessed by X and Y, S may be given one or more topologies, and in some cases itself possesses an additional structure. Many questions of functional analysis, for example, are most conveniently expressed in terms of the topology or metric on families of functions on the real line (Is a periodic function "equal" to its Fourier series? That's asking whether the original function lies in the closure of the sequence of partial sums, which depends on the topology.) Thus we see in topology some treatments of the Arzelà-Ascoli theorem and the Stone-Weierstrass theorem.

Other families of examples include the Euclidean spaces themselves and various subspaces (curves and surfaces, spheres, and so on). Here we may use the combined structures of the Lebesgue measure, the Euclidean metric, and the natural coordinate charts to ask and answer questions of a generally topological nature. Well-known results on Euclidean spaces include the Peano curves covering the square (Hahn-Mazurkiewicz theorem), the Jordan curve theorem, and the Banach-Tarski paradoxical volume-altering decompositions; it's nontrivial even to that R^n and R^m are not homeomorphic unless n=m. Results on spheres include Borsuk-Ulam separation results, Lyusternik-Shnirel´man covering results, and the whole of degree theory (which can count the Fundamental Theorem of Algebra among its consequences!). Much of the topological nature of these spaces is developed in the branches of topology known as Algebraic Topology and Differential Topology.

Other important applications of topology include a number of fixed-point theorems and the topological nature of dynamical systems. The Brouwer fixed-point theorem is perhaps the best-known example, but arguably everything from Newton's method to fractals is a study of the stability and convergence of iterates of functions from a topological space to itself. Clearly many of the most-applicable research in this area is limited to metric spaces or manifolds, but some of the topics may be approached in a fairly general way.

History

See the article on Topology at St Andrews.

Applications and related fields

This section is limited to what is sometimes known as "point-set topology". Some important topological results can be proved with substantial algebraic machinery; particularly strong topological results can be expected of spaces which have additional structure, especially manifolds.

See 55: Algebraic Topology for the definitions, and computations, and applications of fundamental groups, homotopy groups, homology and cohomology. This includes topics in homotopy theory -- studies of spaces in the homotopy category -- whether or not they involve algebraic invariants.

The study of 57: Manifolds and Cell Complexes is roughly the part of the topological category most amenable to nice pictures. This includes differential topology -- what happens when we add differential or other structures? -- actions of groups on spaces, knot theory, low-dimensional topology, and so on.

Simplicial complexes are essentially polyhedra.

Problems specific to Euclidean space may be treated in the geometry pages, particularly if the questions themselves center heavily on the metric.

Aspects of real analysis (differentiation and integration) treated topologically are the purview of global analysis. This includes both the topological study of spaces of maps (e.g. the space of embeddings of one manifold into another) and the study of differential equations and so on with manifolds other than Euclidean space as the domain. In particular, this includes the more detailed studies of dynamical systems, although the related area of fractals is most likely in Measure Theory.

The study of topological vector spaces is treated in detail in functional analysis and related fields such as operator theory. There is also a separate classification for topological groups. In most other cases, "topological xxx theory" is treated as a subset of "xxx theory"

Use of the word "category" (e.g. in the Lyusternik-Shnirel´man theorem) is unrelated to its use in category theory [Schematic of subareas and related areas]

Subfields

Until 1958 an additional classification 56.0X was used for general topology.

Browse all (old) classifications for this area at the AMS.


Textbooks, reference works, and tutorials

There are a number of good textbooks, with varying perspectives on topology.

An online textbook provides a nice development of the basic theory. [Aisling McCluskey and Brian McMaster]

The definitive axiomatic development is of course by Bourbaki, Nicolas, "General topology" (2 volumes), Springer-Verlag, Berlin-New York, 1989. 363+437 pp. ISBN 3-540-19372-3 and 3-540-19374-X (MR90a:54001a,b)

The foundational aspects of the subject are thoroughly reviewed in "Handbook of set-theoretic topology", edited by Kenneth Kunen and Jerry E. Vaughan. North-Holland Publishing Co., Amsterdam-New York, 1984. 1273 pp. ISBN 0-444-86580-2, 85k:54001

There is an excellent, if somewhat dated, collection of "Reviews in Topology" by Norman Steenrod, a sorted collection of the relevant reviews from Math Reviews (1940-1967). Many now-classical results date from that period. Most of the reviews in that collection are of algebraic and differential topology, however.

Some survey articles:

Got a question in this area? Ask a Topologist (bulletin board).

Software and tables

It is difficult to imagine software or tables for topology distinct from those for algebraic topology (e.g. tables of homotopy groups) or what is essentially (numerical) analysis (e.g. implementations of fixed-point routines)! Perhaps the most appropriate candidate would be tables outlining the implications and interplay among the various axioms on topologies; this is essentially the intent of

Other web sites with this focus

Selected topics at this site


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Last modified 2000/01/14 by Dave Rusin. Mail: